A polynomial invariant on all but one variable

The content in today's blog is taken from Jean-Pierre Tignol's Galois' Theory of Algebraic Equations.

Lemma:

Let g be a polynomial in n indeterminates x₁, ..., x_n over some field K.

Let g be invariant under every permutation of x₂, ..., x_n

Then:

g can be written as a polynomial in x₁ and the elementary symmetric polynomials s₁, ..., s_n-1 in x₁, ..., x_n

Proof:

(1) We can view g as a polynomial in x₂, ..., x_n with coefficients in K[x₁].

(2) Using Waring's Method [see Theorem 4, here], we know that g can be written as a polynomial in the elementary symmetric polynomials s'₁, ..., s'_n-1 in x₂, ..., x_n with coefficients in K[x₁]

(3) Therefore, there exists a polynomial g' such that:

g(x₁, ..., x_n) = g'(x₁, s'₁, ..., s'_n-1)

where:

s'₁ = x₂ + ... + x_n

s'₂ = x₂x₃ + ... + x_n-1x_n

...

s'_n-1 = x₂*...*x_n

(4) To complete the proof, we need to show that s'₁, s'₂, ..., s'_n-1 can be restated in terms of s₁, s₂, ..., s_n where:

s₁ = x₁ + ... + x_n

s₂ = x₁x₂ + ... + x_n-1x_n

...

s_n = x₁*...*x_n

(5) We know that for any given polynomial [see Theorem 1, here]:

(X - x₁)*...*(X - x_n) = Xⁿ - s₁X^n-1 + ... + (-1)ⁿs_n

(6) Now, we can use the same principle to get:

(X - x₂)*...*(X - x_n) = X^n-1 - s'₁X^n-2 + ... + (-1)^n-1s'_n-1

(7) Multiplying the above equation by (X - x₁) gives us:

(X - x₁)*...*(X - x_n) = (X - x₁)X^n-1 - (X - x₁)s'₁X^n-2 + ... + (X - x₁)(-1)^n-1s'_n-1 =

Xⁿ - (x₁+s'₁)X^n-1 + (x₁s'₁ + s'₂)X^n-2 - (x₁s'₂ + s'₃)X^n-3 + ... + (-1)ⁿ(x₁s'_n-1)

(8) Combining step #5 and step #7 gives us:

s₁ = x₁ + s'₁

so that:

s'₁ = s₁ - x₁


s₂ = x₁s'₁ + s'₂

so that:

s'₂ = s₂ - x₁s'₁ = s₂ - x₁(s₁ - x₁) = s₂ - x₁s₁ + x₁²

and so on...

(9) Since we can subtitute all values s'_i in terms of x₁ and s₁, ..., s_n, we can use the equation in step #3 to get:

g(x₁, ..., x_n) = g'(x₁, s'₁, ..., s'_n-1) = g'(x₁, s₁ - x₁, s₂ - s₁x₁ + x₁², ... )

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001